1) In the BEGINNING, i.e., with this new fellow, I think I get nervous/worried I am not likable enough and/or he won't be interested so go straight into anxious - checking email, worrying and wondering about him, overthinking my communications, etc . Generally, I get ahead of myself and feel anxious.
2) Then once he (or anyone) gets interested enough to really pursue me (or even just give the appearance of that), I get scared he wants too much and go into avoidant.
I would like to break both sides of this cycle.
Hi ReclaimingMyLife, I just got the e-book of
Attached from the library!

I find that different sources of information about adult attachment styles describe them differently (for instance some recognize a style called "disorganized" and some don't, some make a distinction between avoidant-fearful and avoidant-dismissive, etc.) They also have different things to say about how to change your attachment style as an adult. So I think it's helpful to read multiple sources.
I have seen "anxious" attachment style also referred to as "ambivalent", so I'm wondering if you have an anxious attachment style and at least some of what you describe is accounted for by that? Parents of anxiously attached children are said to be inconsistently responsive (sometimes they meet the child's needs, sometimes they don't) as opposed to consistently unresponsive parenting which correlates with an avoidant attachment style in the child.
Harville Hendrix talks about attachment style in
Keeping the Love You Find, here's a quote: "A clinging child has a highly ambivalent relationship to his mother. Tormented by her unpredictable availability, he is simultaneously addicted to getting her attention and finding a way to get her to respond; at the same time he is angry that his needs aren't being met. He spends half his time crying and holding on, to keep his unpredictable mother by his side, the other half being rejecting, pushing his mother away, even as she is being affectionate. The infant is in a dilemma because the object of pain and pleasure is the same. [... .] Because he cannot live in an inconsistently supportive environment and tolerate for long the consequent negative feelings its insecurity stimulates, he develops an ambivalent defensive structure, alternately clinging and pushing away, to ward off these incapacitating feelings."
I've been skimming through a few chapters of
Attached and one of the things it says for people who are insecurely attached is that they can create for themselves a "working model" of secure attachment, based on times when others have responded to them in a secure way, or how a secure couple they know treat each other, or even relationships with pets.